For a superconducting wedge-like sample with angle alpha, the superconducting state is described and the phase boundaries are analyzed in the framework of the variational approach to the linearized Ginzburg-Landau equation. The nucleation of the superconducting state is shown to be substantially enhanced in the vicinity of the wedge's edge, where the local surface critical field H-c3 exceeds the bulk upper critical field H-c2 by the factor eta(alpha) = H-c3/H-c2 depending on alpha. The enhancement factor eta(alpha) is 1.69 for alpha = pi and is increased up to 3.79 for alpha = 0.44 pi.